Properties

Label 2-1110-5.4-c1-0-7
Degree $2$
Conductor $1110$
Sign $0.100 - 0.994i$
Analytic cond. $8.86339$
Root an. cond. $2.97714$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + i·3-s − 4-s + (2.22 + 0.224i)5-s + 6-s + 1.44i·7-s + i·8-s − 9-s + (0.224 − 2.22i)10-s − 3.44·11-s i·12-s + 5.44i·13-s + 1.44·14-s + (−0.224 + 2.22i)15-s + 16-s + 1.44i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + (0.994 + 0.100i)5-s + 0.408·6-s + 0.547i·7-s + 0.353i·8-s − 0.333·9-s + (0.0710 − 0.703i)10-s − 1.04·11-s − 0.288i·12-s + 1.51i·13-s + 0.387·14-s + (−0.0580 + 0.574i)15-s + 0.250·16-s + 0.351i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1110\)    =    \(2 \cdot 3 \cdot 5 \cdot 37\)
Sign: $0.100 - 0.994i$
Analytic conductor: \(8.86339\)
Root analytic conductor: \(2.97714\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1110} (889, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1110,\ (\ :1/2),\ 0.100 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.254714502\)
\(L(\frac12)\) \(\approx\) \(1.254714502\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 - iT \)
5 \( 1 + (-2.22 - 0.224i)T \)
37 \( 1 - iT \)
good7 \( 1 - 1.44iT - 7T^{2} \)
11 \( 1 + 3.44T + 11T^{2} \)
13 \( 1 - 5.44iT - 13T^{2} \)
17 \( 1 - 1.44iT - 17T^{2} \)
19 \( 1 + 3T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 + 8.44T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
41 \( 1 - 4.44T + 41T^{2} \)
43 \( 1 - 2.89iT - 43T^{2} \)
47 \( 1 - 1.55iT - 47T^{2} \)
53 \( 1 + 3iT - 53T^{2} \)
59 \( 1 + 11.3T + 59T^{2} \)
61 \( 1 + 5.10T + 61T^{2} \)
67 \( 1 - 2.89iT - 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 10.7iT - 73T^{2} \)
79 \( 1 - 1.34T + 79T^{2} \)
83 \( 1 - 3.44iT - 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 - 11.3iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.02677981413070251003614906579, −9.273762996154409955796283754657, −8.882384828724012070007640826905, −7.71866715957542341571894852715, −6.44613257633141442996200539551, −5.62110540793100754369842636544, −4.82861739546831656081622888815, −3.80956787097617649081673133121, −2.55712830449778924058979814898, −1.84290282034480781219204791631, 0.52247514974945832890410220973, 2.12073114387487278879960720074, 3.28871522073292721679674528602, 4.78312513816961127053228106504, 5.60701917596036547908785574005, 6.15575314427819058113314936972, 7.31669869608471295692916382367, 7.75593436364742131113924379821, 8.702446800693879291173115792689, 9.551197416125337822330314654644

Graph of the $Z$-function along the critical line